1

Discrete-Time Ping-pong Optimized Pulse

Shaping-OFDM (POPS-OFDM) Operating on

Time and Frequency Dispersive Channels

for 5G SystemsZeineb Hraiech, Mohamed Siala and Fatma Abdelkefi

MEDIATRON Laboratory, SUP’COM, University of Carthage, Tunisia

Email: zeineb.hraiech, mohamed.siala, fatma.abdelkefi@supcom.tn

Abstract—The Fourth Generation (4G) of mobile communication systems was optimized to offer high data rates with high terminal

mobility by ensuring strict synchronism and perfect orthogonality. However, the trend for novel applications, that had not been feasible a

few years back, reveals major limits of this strict synchronism and imposes new challenges and severe requirements. Among those, the

sporadic access generated by Machine-Type Communication (MTC) and the exchange of small data packets with small payloads,

respecting perfect synchronization procedure, imposes the use of large overhead compared to the useful data and hence lead to a

significant system performance degradation. As a consequence, MTC nodes need to be coarsely synchronized to reduce the signaling

load while reducing the round-trip delay. However, coarse synchronization can dramatically damage the waveforms orthogonality in the

Orthogonal Frequency Division Multiplexing (OFDM) signals, which results in oppressive Inter-Carrier Interference (ICI) and

Inter-Symbol Interference (ISI). Moreover, in 4G, tremendous efforts must be spent to enhance system performance under strict

synchronism in collaborative schemes, such as Coordinated Multi-Point (CoMP). As a consequence, the use of non-orthogonal

waveforms becomes further essential in order to meet the upcoming requirements. In this context, we propose here a novel waveform

construction, referred to Ping-pong Optimized Pulse Shaping-OFDM (POPS-OFDM), which is believed to be an attractive candidate for

the optimization of the radio interface of next 5G mobile communication systems. Through a maximization of the Signal to Interference

plus Noise Ratio (SINR), this approach allows optimal and straightforward waveform design for multicarrier systems at the Transmitter

(Tx) and Receiver (Rx) sides. Furthermore, the optimized waveforms at both Tx/Rx sides have the advantage to be adapted to the

channel propagation conditions and the impairments caused by strict synchronism relaxation for reduced synchronization overhead.

Hence, this straightforward optimization results in spectacular performance enhancement compared to classical multicarrier systems

using conventional waveforms. In this paper, we analyze several characteristics of the proposed waveforms and shed light on relevant

features, which make it a powerful candidate for the design of 5G system radio interface waveforms.

Index Terms—POPS-OFDM, MTC, 5G, Asynchronism, Optimized Waveforms, Out-of-Band (OOB) Emissions, Inter-Carrier

Interference (ICI), Inter-Symbol Interference (ISI), Signal to Interference plus Noise Ratio (SINR)

F

1 INTRODUCTION

R ECENTLY, a potential research confirms that transition

to the next Fifth Generation (5G) of mobile communi-

cation systems becomes further essential in order to meet

the future communication services and needs. First and

foremost, the trend to offer novel applications of wireless

cellular systems that are expected to be feasible by 2020

certainly plays a key role in future communications drivers

and imposes new challenges. Among those applications, the

Tactile Internet [1-3], which comprises real-time applications

with extremely low latency requirements, imposes a time

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budget on the physical layer below 100 µs [3]. As a con-

sequence, the exchange of small data packets with small

payloads, respecting a perfect synchronization procedure,

imposes an intensive exchange of signaling messages such

as synchronization and channel estimation pilot sequences.

Hence, it incurs a large latency which leads to a significant

system performance degradation in terms of efficient use

of energy and radio resources. Coarse synchronization can

solve the problem and reduce the signaling load while

reducing the round-trip delay. In addition to the latter, the

Internet of Things (IoT) [5], which generates sporadic access,

constitutes another significant challenge research domain [1-

4]. Besides a scalability problem, MTC devices need to be

coarsely synchronized to guarantee long lifetimes. However,

strict synchronism relaxation can dramatically damage the

waveforms orthogonality in the Orthogonal Frequency Di-

vision Multiplexing (OFDM) signals, on which many recent

wireless standards rely [10], since it results in oppressive

Inter-Carrier Interference (ICI) and Inter-Symbol Interfer-

ence (ISI). In addition to that, in 4G, several transmission

schemes and applications require strict obedience to syn-

chronism procedure. For example, evolved Nodes B (eNBs)

transmitting Multimedia Broadcast over Single Frequency

Network (MBSFN), should guarantee perfect synchronism

to achieve high operating SNR. Hence, in order to compen-

sate the differences of the propagation delays between eNBs

transmitting MBFSN, the adjunction of extended Cyclic

Prefix (CP), which is greater than the CP used by default for

the channel equalization in the frequency domain, further

constrain the bandwidth efficiency [11], [12] and reduce

data capacity. Also, tremendous efforts must be spent to

enhance system performance under strict synchronism in

collaborative schemes, such as Coordinated Multi Point

(CoMP).

In addition to its sensitivity to synchronism and wave-

form orthogonality, several research studies highlighted a

set of drawbacks of OFDM transmission, including mainly

the out-of-band emissions, which can result in an im-

portant interference level between systems using adjacent

bands. Moreover, Carrier Frequency Offset (CFO), which

requires sophisticated synchronization mechanisms to guar-

antee that the orthogonality is not affected [13], is one

of most well-known disturbances for OFDM. As a con-

sequence, 4G systems can’t match the needs cited above,

since they require tight synchronization and incur high

latencies. Therefore, a non-orthogonal future wireless multi-

carrier scheme with well-localized waveforms in time and

frequency domains would represent an interesting candi-

date to be used for the 5G systems. These novel waveforms

should be robust to relaxed time/frequency synchronization

requirements and imperfect channel state information.

Besides wireless radio frequency applications stated above,

many works shed lights on underwater acoustic applica-

tions such as submarine multimedia surveillance, undersea

explorations, video-assisted navigation and environmental

monitoring [8] as an actual and important applications

domain. However, the underwater acoustic channels are

generally recognized as one of the most difficult commu-

nication media in use today [8], [9]. The acoustic propaga-

tion suffers from severe transmission losses, time-varying

multipath propagation, high Doppler spreads, and high

propagation delays [8], [9]. Thus, in addition to the relaxed

synchronism and imperfect orthogonality requirements, this

area of applications requires the design of novel waveforms

to reduce ISI and ICI and also minimize energy spreading.

Several solutions were proposed in the literature to miti-

gate ICI and ISI, when the propagation channel is doubly

dispersive and the obedience to strict synchronism and

perfect orthogonality are challenged. One of the envisaged

solutions is referred to as Generalized Frequency Division

Multiplexing (GFDM) [11], [18], which is a new concept

for flexible multi-carrier transmission. It offers a low out-

of-band radiation of the transmitted signal compared to

OFDM, due to an adjustable pulse shaping filter that is

applied to the individual subcarriers. As such, it strictly

avoids harmful interference to legacy TV signals, allowing

to opportunistically exploiting spectrum white spaces for

wireless data communications. It also features a block-

based transmission, using CP insertion, possible window-

ing for out-of-band (OOB) spectrum leakage, and efficient

FFT-based equalization. One of the major drawbacks of

GFDM is the requirement to operate on a non-time selec-

3

tive (frequency-dispersive) channel within each transmitted

block and to guarantee perfect frequency synchronization

at the receiver. Unfortunately, for a given CP insertion

overhead, the aggregation of a significant number of trans-

mitted symbols to increase spectrum efficiency makes these

assumptions unrealistic or difficult to meet. Another major

drawback is due to the sampled nature of GFDM and the

use of all potential transmit subcarriers to preserve spec-

trum efficiency, which breaks waveform spectrum shaping

because of spectrum folding. One of the proposed ways to

reduce OOB power emission is to apply some kind empirical

filtering through a multiplication of each transmitted block

by a smooth windowing.

Another serious multiple radio access candidate under

consideration for 5G is Filter Bank Multi-Carrier (FBMC)

[19], [20]. It has many advantages over OFDM, such as

having much better control of the out-of-band radiations

due to the frequency localized shaping pulses. This scheme

discards the concept of CP and relies on a per-subcarrier

equalization to combat ISI and hence improves the spectral

efficiency [21]. However, the low latency requirements are

not guaranteed by FBMC due to the use of long filter

lengths [11], in addition to not guaranteed robustness to

time synchronization imperfections.

Furthermore, a Universal Filtered Multi-Carrier (UFMC)

[22], [3] approach was introduced as an alternative to OFDM

where a filtering operation is applied to a group of con-

secutive subcarriers to reduce the OOB emissions. Here

too, UFMC does not require the use of CP, which makes

UFMC more sensitive to small time misalignment than CP-

OFDM [22]. Hence, it will not be suitable for applications

which need a relaxed time/frequency synchronization re-

quirements.

In the same context, a dynamic waveform construction

referred as Ping-pong Optimized Pulse Shaping-OFDM

(POPS-OFDM) [6] was introduced as an attractive candidate

for the physical layer of 5G systems in order to eliminate the

sensitivity to orthogonality and synchronism constraints.

This innovative approach banks on a non-orthogonal wire-

less multi-carrier scheme which allows the design of well-

localized waveforms in both time and frequency domains

[7]. Through an iterative maximization of the Signal to

Interference plus Noise Ratio (SINR), POPS-OFDM deduces

the optimized waveforms at Tx/Rx sides. In addition to

ICI/ISI mitigation, these waveforms deal efficiently with the

problem of the small data packets introduced by the IoT and

the MTC in future wireless communication systems, since

they are robust against time/frequency synchronization er-

rors. In this paper, we analyze several characteristics of the

proposed waveforms and shed light on relevant features.

We note that the conference versions of this paper [6] and

[7] contain parts of the results presented herein. However,

we believe that the actual version is a substantial extension

of the conference versions, where we detail in a precise way

the different concepts. We highlight below the main extra

contributions beyond the conference versions:

• We present a discrete-Time POPS-OFDM version in

a general context, without any constraint on the

choice of the channel propagation. However, to make

the simulations tractable, we consider a radio chan-

nel where the scattering function has a multipath

power profile with an exponential truncated decay-

ing model and classical Doppler spectrum (Section

6).

• We evaluate the complexity of POPS-OFDM imple-

mentation through a careful presentation of its un-

derlying methodology.

• We derive an upper bound on the SINR of POPS-

OFDM algorithm and the exact SINR expression of

the conventional OFDM systems (Sections 5 and 4.4.1

respectively).

• We test the performance of POPS-OFDM for different

Transmitter/Receiver (Tx/Rx) pulse shape durations

and we highlight the inversion properties in time and

hence characterize the properties of the optimization

solution, once it was unique.

• We draw a discussion on the correlation noise and

the careful choice of the couple of transmitted and

received waveforms or their temporal inverses with

exchange of Tx/Rx roles, to reduce noise correlation.

• We precise carefully the treatment done in cases

4

of classical OFDM, the discovery of the duality of

CP-OFDM and ZP-OFDM systems and the fact that

CP-OFDM has a zero correlation between the noise

samples at the receiver, while ZP-OFDM shows a

frequency correlation between the subcarriers of one

OFDM symbol which tends, in the absence of a

good interleaving, to reduce correction capacity of

the used error correcting code.

• We assess POPS-OFDM sensitivity to an estimation

error of the channel spread factor and its robustness

against the time and frequency synchronization er-

rors.

The rest of this paper is organized as follows. In Sec-

tion III, we present our multicarrier system model and we

specify its transmitter and receiver blocks. We also detail

the propagation channel models that will be used in this

paper. In Section IV, we focus on the derivation of the

SINR expression and describe the iterative POPS-OFDM

technique for waveform design. In Section V, we illustrate

the obtained optimization numerical results and highlight

the efficiency of the proposed POPS-OFDM algorithm. Sec-

tion VI will be dedicated to evaluate numerically the opti-

mization technique in terms of robustness against time and

frequency synchronization errors and its sensibility to wave-

form initializations. Finally, section VII draws conclusions

and perspectives of our work.

2 NOTATIONS

Notation Meaning

<z = x The real part of the complex num-

ber z = x+ iy

E The expectation operator

V = [Vpq]p,q∈Z The doubly underlined V refers to

matrix V with (p, q)th entry Vpq

Φνk

Hermitian matrix with (p, q)th entry

ej2πνkTs(q−p)

V = [Vq]q∈Z Underlined V refers to vector V

with qth entry Vq

I Identity matrix, with ones in the

main diagonal and zero elsewhere

σp(V ) Time shift by p samples of vector

V = [Vq]q∈Z, i.e, σp(V ) = [Vq−p]q∈Z

$(V ) Temporal inversion of vector V , i.e,

$(V ) = [V−q]q∈Z

δ(i− j) =

1 i = j

0 i 6= jKroncker delta function

.H Hermitien transpose operator

.T Transpose operator

.∗ Element-wise conjugation operator

Component-wise product of two

vectors or matrices

⊗ Kronecker product of two vectors

(X ⊗ V = [XqV ]q∈Z)

< U, V > Hermitian scalar product of U and

V

||V || Norm of a vector V , ||V || =√< V , V >

3 SYSTEM MODEL

This section provides preliminary concepts and notations

related to the considered multicarrier scheme and the chan-

nel assumptions and model. Furthermore, we consider their

discrete time version in order to simplify the implementa-

tion of this technique and hence, the theoretical derivations

that will be investigated.

3.1 OFDM Transmitter and receiver blocks

We denote by T the OFDM symbol duration and by F the

frequency spacing between two adjacent subcarriers. The

5

transmitted signal is sampled at a sampling rate Rs = 1Ts

,

where Ts is the sampling period. We choose the symbol

duration as an integer multiple of the sampling period, i.e.

T = NTs , where N ∈ N∗. We also choose the subcarrier

spacing inverse 1/F as an integer multiple of the sampling

period, i.e. 1/F = QTs , where Q ∈ N∗.

The time-frequency lattice density of the studied OFDM

system, which is always taken below unity to account for the

equivalent effect of guard interval insertion in conventional

OFDM, is defined as

∆ =1

FT

Hence

∆ = QTs1

NTs=Q

N≤ 1,

which means that Q is always smaller than or equal to

N and that the time-frequency lattice density is always

rational. The difference (N−Q)Ts is equivalent to the notion

of guard interval in conventional OFDM.

Because of sampling, the total spanned bandwidth is equal

to 1Ts

and therefore the number of subcarriers is finite and

equal to (1/Ts)/F = Q. Therefore, adopting [0, 1Ts

) as

the spanned frequency band, the subcarrier frequencies are

given by mF = m/QTs, m = 0, 1, .., Q− 1.

The sampled version of the transmitted signal is represented

by the infinite vector

e = (...., e−2, e−1, e0, e1, .....)T = [eq]q∈Z

where eq is the transmitted signal sample at time qTs.

The transmitted signal can be written as

e =∑mn

amn ϕmn (1)

where amn is the transmitted symbol at time nT and fre-

quency mF , and

ϕmn

= [ϕ(q − nN)ej2πmqQ ]q∈Z

is the vector used for the transmission of symbol amn, which

results from a time shift of nNTs = nT and a frequency

shift of mF = m/QTs of the transmission prototype vector

ϕ. Also, we denote by DϕT the support duration of the

transmitter waveforms, where Dϕ ∈ N∗. We suppose that

this duration is a finite number in order to reduce the latency

and complexity. We denote by E = E[|amn|2] ||ϕ||2, the

average energy of the transmitted symbol at time nT and

frequency mF , where ‖ϕ‖2 =∑q∈Z |ϕq|2.

Assuming a linear time-varying multipath channel h(p, q),

with p and q standing respectively for the normalized time

delay and the normalized observation time, the sampled

version of the received signal, r = [rq]q∈Z, has the following

expression:

r =∑mn

amnϕmn + n (2)

where

ϕmn

=

[∑p

h(p, q)ϕmn(q − p)]q∈Z

(3)

is the channel-distorted version, ϕmn

, of ϕmn

, and n is

a discrete-time complex additive white Gaussian noise

(AWGN), the samples of which are centered, uncorrelated

with common variance N0, where N0 is the two-sided

spectral density of the original continuous-time noise. To

simplify the derivation and make it tractable, while keeping

the presentation general, we consider a channel with a finite

number, K , of paths, with channel impulse response

h(p, q) =K−1∑k=0

hkej2πνkTsqδ(p− pk),

where hk, νk and pk are respectively the amplitude, the

Doppler frequency and the time delay of the kth path. The

paths amplitudes hk, k = 0..K − 1, are assumed to be cen-

tered and decorrelated random complex Gaussian variables

with average powers πk = E[|hk|2], k = 0..K−1. In order to

make the simulation tractable, the paths amplitudes hk are

assumed to be i.i.d. complex Gaussian variables with zero

mean and∑K−1k=0 πk = 1.

The channel scattering function is therefore given by

S(p, ν) =K−1∑k=0

πkδ(p− pk)δ(ν − νk).

At the receiver side, the decision variable on symbol akl is

given by

Λkl =⟨ψkl, r⟩

= ψHklr (4)

6

where

ψkl

= [ψ(q − lN)ej2πkqQ ]q∈Z

is the time and frequency shifted version, by lT in time and

kF in frequency, of the reception prototype vector ψ used

for the demodulation of akl. Here too, we denote by Dψ , the

support duration of the receiver waveform.

Here, we relax the constraint on the Tx/Rx pulses to be

identical, leading to a greater flexibility in the optimization

process and to an additional increase in the achievable SINR.

At this point, it is worth mentioning that OFDM with CP

or Zero Padding (ZP), also use different waveforms at the

Tx/Rx to account for CP and ZP insertion at the transmitter

respectively.

4 POPS-OFDM ALGORITHM

In this section, we will present the purpose of POPS OFDM

algorithm which aims to design optimal waveforms at the

Tx/Rx sides through SINR maximization, for fixed channel

and synchronization imperfections statistics. Since POPS-

OFDM is an iterative algorithm, it alternates between an op-

timization of the receiver waveform ψ, for a given transmit

waveform ϕ, and an optimization of the transmit waveform

ϕ for a given receive waveform ψ.

Without loss of generality, we will focus on the evaluation

of the signal to interference plus noise ratio (SINR) for

symbol a00. This SINR will be exactly the same for all other

transmitted symbols. Referring to (4), the decision variable

on a00 can be expanded into three additive terms, as

Λ00 = a00

⟨ψ

00, ϕ

00

⟩︸ ︷︷ ︸

U00

+∑

(m,n)6=(0,0)

amn⟨ψ

00, ϕ

mn

⟩︸ ︷︷ ︸

I00

+⟨ψ

00, n⟩

︸ ︷︷ ︸N00

The first term, U00, is the useful part in the decision variable.

Its power represents the useful signal power in the SINR.

The second term, I00, is the inter-symbol interference, , ac-

counting for ISI and ICI, and the last term, N00, is the noise

term. Their respective powers represent the interference and

the noise powers in the SINR.

4.1 Average Useful Power

The useful term in the decision variable on a00 is given by

U00 = a00 < ψ00, ϕ

00>. For a given realization of the

channel, the average power of the useful terms is given by

PhS = E||ϕ||2 | < ψ

00, ϕ

00> |2. Therefore, the average of

the useful power over channel realizations is PS = E[PhS ].

Under the notions cited above, we deduce that

PS = EψHKS

ϕ

S(p,ν)ψ

||ϕ||2, (5)

where we define the useful signal Kernel matrix as

KSϕ

S(p,ν) =K−1∑k=0

πk (σpk(ϕ00

)σpk(ϕ00

)H) Φνk

=K−1∑k=0

πk (σpk(ϕ)σpk(ϕ)H) Φνk

(6)

Given that PS is a positive entity quantity, we can state

that the Kernel matrix is a positive Hermitian matrix. Since

POPS-OFDM is an iterative algorithm, where ψ and ϕ have

to exchange alternately their roles, it is essential to introduce

this propriety relating KSϕ

S(p,ν) and KSψ

S(−p,−ν):

ϕHKSψ

S(p,ν)ϕ = ψHKSϕ

S(−p,−ν)ψ. (7)

These equalities say that the useful signal power can be

expressed as a quadratic form on ψ for a given ϕ and vice

versa, given propagation channel and synchronization error

statistics, summarized in the scattering function S.

4.2 Average Interference Power

The interference term within the decision variable Λ00,

given by I00 =∑

(m,n)6=(0,0) amn < ψ00, ϕ

mn>, results

from the contribution of all other transmitted symbols amn

such that (m,n) 6= (0, 0). The mean power of PhI , over

channel realizations, is given by

PI = E[PhI ] =E

||ϕ||2∑

(m,n)6=(0,0)

E[| < ψ00, ϕ

mn> |2].

By reiterating the same derivation as the one in Section 4.1,

we find that:

PI = EψHKI

ϕ

S(p,ν)ψ

||ϕ||2. (8)

7

where the interference Kernel matrix is expressed as

KIϕ

S(p,ν) =K−1∑k=0

πk

∑(m,n)6=(0,0)

σpk(ϕmn

)σpk(ϕmn

)H

Φνk.

(9)

Since PI is always positive, the interference kernel KIϕ

S(p,ν),

is also Hermitien positive semidefinite matrix.

Here too, it is important to highlight an important property

that both KIϕ

S(p,ν) and KIψ

S(−p,−ν) do verify, namely

ψHKIϕ

S(p,ν)ψ = ϕHKIψ

S(−p,−ν)ϕ.

ϕHKIψ

S(p,ν)ϕ = ψHKIϕ

S(−p,−ν)ψ. (10)

This property is very important for the following part of the

paper: given any arbitrary choice of the receiver prototype

vector ψ, we can optimize the choice of the transmitter

prototype vector ϕ through a maximization of the SINR.

4.3 Average Noise Power

Foremost, we try to express the noise correlation between

the two samples Nmn and Nkl, where the noise is white:

E[N∗mnNkl] = E[< ψmn, r >∗ < ψ

kl, r >]

= ψHklE[n nH ] ψ

mn

= N0 ψH

klψmn

= N0 < ψkl, ψ

mn> (11)

We remark here that the noise correlation between samples

depend only on the received pulse ψ and not to the transmit-

ted pulse ϕ. Taking (k, l) = (m,n) in the previous equation,

we find ..., we find that the average power of noise term,

Nkl, to be equal to

PN = N0

∥∥∥ψkl

∥∥∥2

= N0

∥∥ψ∥∥2. (12)

4.4 SINR Expression

Using the obtained expressions of the useful power PS , the

interference power PI and the noise power PN , We can

express the SINR as the following expression:

SINR =PS

PI + PN=

ψH KSϕ

S(p,ν) ψ

ψH (KIϕ

S(p,ν) +||ϕ||2SNR I) ψ

, (13)

where SNR = EN0

is the Signal to Noise Ratio. This

expression is valid for a general channel model without

any limitations. This equation is useful in the optimization

process in order to determine the received waveform, given

a particular choice of the transmitted waveform at the

beginning of the optimization process.

We notice that, by interchanging ϕ and ψ roles, i.e. by letting

ψ to be the transmitted waveform and ϕ to be the received

waveform, the resulting SINR remains unchanged. In fact,

using the previous identities in (13), we can write:

SINR =PS

PI + PN=

ϕH KSψ

S(−p,−ν) ϕ

ϕH (KIψ

S(−p,−ν) +||ϕ||2SNR I) ϕ

, (14)

This equation allows the design of the optimized transmit-

ted waveform, given a particular choice of the received

waveform. Also, while maintaining the initial scattering

function, S(p, ν), and by interchanging the transmitted and

received waveforms and taking their time inversed versions,

$(ψ),$(ϕ), we can express the SINR as:

SINR =PS

PI + PN=

$(ϕ)HKS

$(ψ)

S(p,ν) $(ϕ)

$(ϕ)H

(KI$(ψ)

S(p,ν) +||$(ψ)||2SNR I) $(ϕ)

,

(15)

One of the consequences of this expression is to simplify

the optimization code by keeping the same code for ψ

optimization given ϕ, since we preserve the scattering

function without any alteration. We only need to plug the

time reverse of ψ in the kernel expressions to be able to

obtained the corresponding time reverse of the optimum

received waveform. Another consequences is that if a couple

of transmitted/received waveforms (ϕ, ψ) achieve a given

SINR, then the interchange of their roles, while taking their

time reverse versions, lead to the same SINR. Last but not

less important, in the maximization process, if the optimal

couple (ϕ,ψ), maximizing the SINR, comes to be unique,

then we certainly have $(ϕ) = ψ, which means that the

transmitted waveform ϕ and the received waveform ψ are

reverse-time of each other. As we will see later, in this

specific case, there is no way to reduce the noise correlation

at the receiver by interchanging ϕ and ψ.

8

4.4.1 SINR expression for the conventional OFDM system

In the previous sections, we derived the SINR expression

without any type’s constraints for the Tx/Rx waveforms.

Hence, (13) is valid for whatever the transmitted and re-

ceived waveforms. In particular, it is valid for the con-

ventional OFDM system. We remind that the transmitted

and received pulses for the OFDM system are expressed as

follows respectively ϕcv = [ϕcvq ]q , where

ϕcvq =

1√N

if q = −(N −Q) · · · 0 · · · (Q− 1),

0 else,(16)

and ψcv = [ψcvq ]q , where

ψcvq

=

1√Q

if q = 0 · · · (Q− 1),

0 else.(17)

As is known, the received waveforms of the CP-OFDM

system, when we discard the CP, constitute an orthonor-

mal base. Similarly, it is the transmitted waveforms in the

ZP-OFDM system, once we discard the ZP, constitute an

orthonormal base. As a consequence, we can simply prove

that the sum of the useful power (5) and the interference

power (8) is equal to:

P cvS + P cvI =E

||ϕcv||2∑

(m=0..Q−1,n)

E[| < ψcv00, ϕcv

mn> |2]

(18)

= EQ

N(19)

In order to calculate the SINR (SINRcv) for the conven-

tional OFDM system , we need to determine the useful

power, P cvS . By injecting (16) in the useful Kernel (6), we

obtain:

KSϕcv

S(p,ν)=

1N · · · ∑K−1

k=0 γk(N−pk−1)∑K−1k=0 γk(−1) · · · ∑K−1

k=0 γk(N−pk−2)

.... . . · · ·∑K−1

k=0 γk(−N+pk+1) · · · 1N

(20)

where

γk(x) =πkNej2πνkTsx. (21)

Hence, based on (19) and (20), we can deduce the conven-

tional SINR for OFDM system, which is equal to

SINRcv =P cvSE

QN − (

P cvSE ) + 1

SNR

(22)

where the useful power, P cvS , using (21), is expressed as

follows:

P cvS =

E∑K−1k=0 γk(0)+E

∑K−1k=0

∑Q−1r=1

2(Q−r)Q <γk(r)

if maxk=0..K−1

pk ≤ N −Q

E∑K−1k=0

N−pkQ γk(0)+

∑K−1k=0

(∑N−pk−1

r=12E(N−pk−r)

Q <γk(r))

else

(23)

4.4.2 Noise Correlation

The couples (ϕ,ψ) and ($(ϕ), $(ψ)) provide the same SINR

of the foregoing and in this sense, they are duals of each

other. However, it is always the waveform at the reception

that determines the noise correlation in time and frequency

which is tainting the decision variables on adjacent sym-

bols in time or/and frequency. Thus, generally, and this

is also the case, CP-OFDM and its dual ZP-OFDM, the

correlation provided by the couple ($(ϕ), $(ψ)) through

ψ is generally different from the correlation provided by

the dual ($(ϕ), $(ψ)) through $(ϕ). In light of the duality

characteristic stated above, we can notice that CP-OFDM

and ZP-OFDM are duals of each other. However, unlike CP-

OFDM, ZP-OFDM induces induces a correlation between

the noise samples in the decision variables.

4.5 Optimization Technique

POPS-OFDM alternates between an optimization of the

transmit waveform ϕ, given the receive waveform ψ and

the optimization of the receive waveform ψ, given the

transmit waveform ϕ). This is the reason why it is called the

Ping-pong Optimized Pulse Shaping (POPS) algorithm. We

remind that POPS-OFDM is an iterative algorithm. Hence,

the choice of the waveform initialization is primordial and

critical to be able to converge to the global maximum and

find the optimal Tx/Rx waveform couple (ψopt, ϕ

opt), that

9

maximizes the SINR. POPS-OFDM is evaluated through Al-

gorithm 1 for a fixed waveform initialization (ϕ(0)) and fixed

channel parameters. More precisely, for the kth iteration, we

have ψ(k−1) available. We start by optimizing ϕ according

to

ϕ(k) = arg maxϕ

ϕH KSψ(k−1)

S(−p,−ν) ϕ

ϕH KINψ(k−1)

S(−p,−ν) ϕ(24)

where KINψ

S(−p,−ν) = KIψ

S(−p,−ν) +||ψ||2

SNR I . Then, given

ϕ(k), we carry an optimization of ψ according to

ψ(k) = arg maxψ

ψH KSϕ(k)

S(p,ν) ψ

ψH KINϕ(k)

S(p,ν) ψ(25)

where KINϕ

S(p,ν) = KIϕ

S(p,ν) +||ϕ||2

SNR I .

4.5.1 First Approach

Since KSϕ

S(p,ν) and KINϕ

S(p,ν) are Hermitian, symmet-

ric, positive and semidefinite, it turns out that our prob-

lem (24) amounts to a maximization of a generalized

Rayleigh quotient [24], which appears in many problems

in engineering and pattern recognition. So, once KINϕ

S(p,ν)

is invertible, our optimization problem becomes a max-

imization one where its solution is the eigenvector of

(KINϕ

S(p,ν))−1 KS

ϕ

S(p,ν) with maximum eigenvalue (see

Algorithm 1).

Algorithm 1 First Approach

Require: Channel parameters (K , pk, νk, Ts, hk), ϕ(0), ε =

10−10, ψ(0) = (0 · · · 0)T , e(ψ) = e(ϕ) = 10, k = 0, SNR

Compute KSϕ(0)

S(p,ν) and KIϕ(0)

S(p,ν)

while e(ψ) > ε or e(ϕ) > ε doKIN

ϕ(k)

S(p,ν) = KIϕ(k)

S(p,ν) +||ϕ(k)||2

SNR I

Compute Φ = (KINϕ(k)

S(p,ν))−1 KS

ϕ(k)

S(p,ν)

Compute [ψ(k), λmax] = eig(Φ)k ← k + 1

Evaluate KIψ(k)

S(−p,−ν), KSψ(k)

S(−p,−ν) and

KINψ(k)

S(−p,−ν) = KIψ(k)

S(−p,−ν) +||ψ(k)||2

SNR I

Compute Θ = (KINψ(k)

S(−p,−ν))−1 KS

ψ(k)

S(−p,−ν)

Compute [ϕ(k), νmax] = eig(Θ)

Evaluate error: e(ψ) = ‖ψ(k) − ψ(k−1)‖ and e(ϕ) =

‖ϕ(k) − ϕ(k−1)‖end while

4.5.2 Second approach

Without loss of generality, through this section, we consider

the optimization of the receive waveform ψ, given the

transmit waveform ϕ.

A possible approach to minimize the SINR consists in min-

imizing the denominator of (13), ψHKIϕ

S(p,ν)ψ, subject to

a fixed useful power, and this can be performed through

the Lagrange multiplier method. This approach is useful to

minimize the normalized interference power for a fixed N0

E

value. In this case our denominator minimization problem

becomes equivalent to consider the following Lagrangian

function:

Qλ,SNRS(p,ν) (ϕ,ψ) = ψH(KIϕ

S(p,ν) − λKSϕ

S(p,ν))ψ (26)

where λ is the Lagrange multiplier that depends on the

Signal to Noise Ratio (SNR) value. To assess the value

around which we should choose the Lagrange multiplier,

we take the gradient of the SINR with respect to ϕ or ψ

.Then, we just make an identification of the obtained terms.

The gradient of the SINR with respect to ψ is given by:

∂SINR

∂ϕ=

−2ψHKSϕ

S(p,ν)ψ

(ψHKIϕ

S(p,ν)ψ + N0

E )2(KI

ϕ

S(p,ν)ψ −1

SINRKS

ϕ

S(p,ν)ψ).

(27)

The vector ψ leading to the optimum SINR corresponds to

a null value of the gradient, i.e.

KIϕ

S(p,ν)ψ −1

SINRKS

ϕ

S(p,ν)ψ = 0. (28)

Similarly, the vector ψ that cancels the auxiliary gradient

function (26) must verify the following equality:

∂

∂ψQλ,SNRS(p,ν) (ϕ,ψ) = 2(KI

ϕ

S(p,ν) − λKSϕ

S(p,ν))ψ = 0. (29)

Referring to expression (28), the Lagrange multiplier λ

should be equal to the inverse of the SINR. The optimization

problem could be solved by the generalized eigenvalue

problem (GEP), since we have to solve (29). As our object

is to maximize the SINR and as 1λ is an eigenvalue of

(KIϕ

S(p,ν),KSϕ

S(p,ν)) in expression (28), the optimum value

λopt = 1SINRmax

corresponds to its maximum eigenvalue

in order to meet our expectations. It results that ψopt

is the

10

eigenvector associated to the smallest eigenvalue λopt of the

GEP (KSϕ

S(p,ν), KIϕ

S(p,ν)) (see Algorithm 2).

Algorithm 2 Second Approach

Require: channel parameters (K , pk, νk, hk, Ts), ϕ(0), ε,ψ(0) = 0, e(ψ) = e(ϕ) = 2, k = 0, SNR

Compute KSϕ(0)

S(p,ν) and KIϕ(0)

S(p,ν)

while e(ψ) > ε or e(ϕ) > ε doλ = eig(KS

ϕ(k)

S(p,ν),KIϕ(k)

S(p,ν))k ← k + 1ψ(k) eigenvector associated to λmin

Evaluate KIψ(k)

S(−p,−ν) and KSψ(k)

S(−p,−ν)

ν = eig(KSψ(k)

S(−p,−ν),KIψ(k)

S(−p,−ν))

ϕ(k) eigenvector associated to νminEvaluate errors: e(ψ) = ‖ψ(k+1) − ψ(k)‖ and e(ϕ) =

‖ϕ(k+1) − ϕ(k)‖end while

We are coming out with a Lagrange multiplier evaluation

that can be characterized as a direct, simple and streamlined

approach.

4.5.3 Third Approach

Another direct optimization method consists in diagonaliz-

ing the SINR denominator of expression (13) and then per-

forming a basis change that will simplify the expression of

this denominator, so that our optimization problem becomes

a maximization matrix that implies finding the eigenvector

of the SINR numerator that corresponds to its maximum

eigenvalue. More precisely, we first introduce the Kernel

function KINϕ

S(p,ν) = KIϕ

S(p,ν) + (N0

E )I . The eigen decom-

position of KINϕ

S(p,ν) is KINϕ

S(p,ν) = U Λ UH , where U

is a unitary matrix, Λ is a diagonal one with non-negative

real values on the diagonal. Then, the SINR denominator

can be written as ψHKINϕ

S(p,ν)ψ = ψHU Λ UHψ = uHu

where u = Λ12UHψ. Since KI

ϕ

S(p,ν) is a positive semidef-

inite matrix, then all the entries of Λ are positive and

greater than N0

E which is larger than or equal to 0. Therefore,

ψ = U Λ−12u and the SINR expression becomes

SINR =uHΦ u

uHu,

where Φ = Λ−12UHKS

ϕ

S(p,ν)U Λ−12 is a positive matrix.

Hence, maximizing the SINR is equivalent determining the

maximum eigenvalue of Φ and its associated eigenvector

umax. Hence, ψopt =U Λ−

12 umax

||U Λ−12 umax||

(see Algorithm 3).

Algorithm 3 Third Approach

Require: channel parameters (K , pk, νk, hk, Ts), ϕ(0), ε,ψ(0) = 0, e(ψ) = e(ϕ) = 2, k = 0, SNR

Compute KSϕ(0)

S(p,ν) and KIϕ(0)

S(p,ν)

while e(ψ) > ε or e(ϕ) > ε doKIN

ϕ(k)

S(p,ν) = KIϕ(k)

S(p,ν) + 1SNRI

Compute [U,Λ] = eig(KINϕ(k)

S(p,ν))

Compute Φ = Λ−12UHKS

ϕ(k)

S(p,ν)U Λ−12

Compute [umax, λmax] = eig(Φ)k ← k + 1

ψ(k) =U Λ−

12 umax

||U Λ−12 umax||

Evaluate KIψ(k)

S(−p,−ν), KSψ(k)

S(−p,−ν) and

KINψ(k)

S(−p,−ν) = KIψ(k)

S(−p,−ν) + 1SNRI

Compute [V ,Σ] = eig(KINψ(k)

S(−p,−ν))

Compute Θ = Σ−12V HKS

ψ(k)

S(−p,−ν)V Σ−12

Compute [vmax, νmax] = eig(Θ)

ϕ(k) =V Σ−

12 vmax

||V Σ−12 vmax||

Evaluate errors e(ψ) = ‖ψ(k+1) − ψ(k)‖ and e(ϕ) =

‖ϕ(k+1) − ϕ(k)‖end while

It is important to note that this third approach is slower than

the second one in terms of necessary compilation resources

and leads approximately to the same performances but more

stable numerically. However, the first approach is the fastest

and the simplest approach compared to the others and leads

to the same performance with more stable computation.

5 OPTIMAL SINR VALUE

As we mentioned before, POPS-OFDM is an iterative algo-

rithm permitting a systematic construction of the optimal

waveforms at Tx/Rx sides. Unfortunately, the function to

be optimized includes several local maxima in addition to

one or more global maxima. As a consequence and cause

of its nature, POPS-OFDM may be trapped in a local maxi-

mum, if the initialization waveform is not chosen carefully,

and hence waveform initializations choice, which will be

discussed in the following section, is very important. In this

context, having an upper bound is beneficial to identify the

waveform initialization that guarantees an optimal wave-

form design with high SINR.

11

To go further in the derivation of the upper bound of the

SINR, we show that we can express the exact SINR as

follows:

SINR =(ϕ⊗ ψ)H A

S(p,ν)(ϕ⊗ ψ)

(ϕ⊗ ψ)H BS(p,ν)

(ϕ⊗ ψ) (30)

where

AS(p,ν)

=K−1∑k=0

Ω(00)k

BS(p,ν)

=∑

(m.n)6=(0,0)

K−1∑k=0

Ω(mn)k

(31)

with

Ω(mn)k

= UTpk+nN

Πmk

Upk+nN

,∀(m,n) ∈ Z, (32)

with

Πmk

= [πk ej2π(νkTs+

mQ )(q−q′)]q,q′∈Z

Ud

=

1 if q mod (d+m×D ×N) = 0

0 else

q,q′∈Z

(33)

Hence, we can conclude the upper bound of the SINR that

POPS-OFDM can reach it without ϕ or ψ initializations and

our problematic will be expressed as follows:

SINR = maxχ

χH AS(p,ν)

χ

χH BS(p,ν)

χ(34)

where χ = ϕ ⊗ ψ = [ϕqψ]q∈Z is the Kronecker product

between ψ and ϕ.

By removing the restriction on χ to be in the form of a

Kronecker product of two vectors, ϕ and ψ, and letting

it to span freely the whole space, we obtain, through a

maximization step, an upper bound (SINR). Since AS(p,ν)

and BS(p,ν)

are symmetric, positive and semidefinite, the

maximization problem in (15) turns out to be a straight-

forward maximization of a generalized Rayleigh quotient.

Hence, the SINR upper bound is the maximum eigenvalue

of B−1S(p,ν)

AS(p,ν)

(see Algorithm 4).

Algorithm 4 : Upper bound of the SINRRequire: Channel parameters (K , pk, νk, Ts, hk), SNR

Compute AS(p,ν)

and BS(p,ν)

Compute ∆ = (BS(p,ν)

)−1 AS(p,ν)

Compute [χ, SINR] = eig(∆)

6 NUMERICAL WAVEFORMS CHARACTERIZATION

We consider a radio mobile channel where the scattering

function S(p, ν) has a multipath power profile with an ex-

ponential truncated decaying model and classical Doppler

spectrum. Let 0 < b < 1 be the decaying factor, such that

the paths powers can be expressed as πk = 1−b1−bK b

k. We

recall that we work with sampled signals which also leads

to use a sampling channel in time domain and therefore the

Doppler spectral density, denoted by α(ν), is periodic in

frequency domain with period 1Ts

. This scattering function

obeys to the Jakes model that is decoupled from the dis-

persion in the time domain denoted β(p). This means that

S(p, ν) = β(p)α(ν), such that β(p) =∑K−1k=0 πkδK(p − pk)

and

α(ν) =

1

πBd1√

1−( 2νBd

)2if |ν| < Bd

2

0 if Bd2 ≤ |ν| ≤1

2Ts

(35)

where Bd is the Doppler spread. Hence, the useful and the

interference Kernel matrices, derived in (6) and (9), will be

expressed respectively as follows:

KSϕ

S(p,ν) =K−1∑k=0

πkσpk(ϕϕH) Φ (36)

KIϕ

S(p,ν) =

(∑n

σnN (K−1∑k=0

πkσpk(ϕϕH)) Ω

)−KSϕS(p,ν)

(37)

where Φ and Ω are the Hermitian matrices for the useful

and the interference kernel matrices expressed respectively

as follows:

Φ = [

∫να(ν)ej2πνTs(q−p)]pq

= [J0(πBdTs(p− q))]pq. (38)

and

Ω = [Ωpq]pq. (39)

12

with

Ωpq =

QJ0(πBdTs(p− q)) if (p− q) mod Q = 0

0 else

(40)

Hence, for the same context, the SINR for the conventional

OFDM will be expressed as follows:

SINRcv =P cvSE

QN − (

P cvSE ) + 1

SNR

(41)

where the useful power (P cvS ), is expressed as follows:

P cvS =

EN [1+

∑Q−1r=1

2(Q−r)Q J0(πBdTsr)] if max

k=0..K−1pk ≤ (N −Q)

EN

[∑K−1k=0

N−pkQ πk+

∑K−1k=0 πk(

∑N−pk−1

r=12(N−pk−r)

Q J0(πBdTsr))]

else

(42)

6.1 POPS-OFDM Implementation Methodology

Through this section, we highlight a very important point

which is the simplicity in the implementation of POPS-

OFDM algorithm. In fact, as we can see in Fig.1(a), the

matrices that depended on the Doppler channel will be com-

puted one time for both useful and interference matrices.

Then, as is depicted in Fig.2, we calculate the dispersion

in the time according to the multipath power profile. After

that, we select the matrix which has the highest energy.

Hence, we can deduce the useful kernel matrix using the

first formula presented in Fig.1(a). Furthermore, we shift the

found matrix according to the normalized symbol duration

N (See Fig.2). Then, we select, as usual, the matrix with

the highest energy to be used in the calculation of the

interference kernel matrix, based on the second formula

depicted in Fig.1(a).

6.1.1 POPS-OFDM with different Tx/Rx Pulse Shape Du-

rations

Many researchers shed lights on the adjacent channel inter-

ference which is caused by both transmitter non-idealities

and imperfect receiver filtering [25]. This type of interfer-

ence need to be reduced because it contributes in network

performance degradation [25], [26]. Mainly due to the trans-

mitter non-linearity, the spectrum mask from transmitter

will leak into adjacent channels. This interference is referred

as the Out-of-Band (OOB) emissions in the frequency do-

main. This is a very important system parameter, since it

is essential for the co-existence of parallel communications

on adjacent channels whether pertaining to the same sys-

tem or to different systems [25]. Hence, in the literature,

engineers define Adjacent Channel Leakage power Ratio

(ACLR) parameter which is the ratio of the transmitted

power to the power measured after a receiver filter in the

adjacent RF channel [25]. ACLR determines the allowed

transmitted power to leak into the first or second neighbor-

ing carrier. Hence, large ACLR will guarantee a reduction

of the adjacent channel interference. Furthermore, in the

receiver side, we have additional interference from adjacent

channels, since the receiver filter cannot be ideal [26]. The

adjacent Channel Selectivity (ACS) parameter is a measure

of the receiver ability to receive a signal at its assigned

channel frequency in the presence of a modulated signal in

the adjacent channel. A poor ACS performance may lead to

dropped calls in certain areas of the cells, also called ‘dead

zones’ [25].

Using a large waveform duration brings this waveform

closer to the ideal and perfect filtering mask, since it in-

creases the ACLR and the ACS in the Tx and Rx sides respec-

tively. However, there is a trade-off between the reduction of

the adjacent channel interference and terminal power con-

sumption and service delay (low latency requirements). We

recall that POPS-OFDM offers the possibility to get flexible

with different Tx/Rx pulse shape durations. In this context,

it comes the idea to investigate POPS-OFDM with malleable

Tx/Rx pulse shapes durations. Hence, the implementation

methodology of the POPS-OFDM where the Tx/Rx pulse

13

(a) Taking into account the lattice periodic structure and repet-itive structure in frequency and the channel Doppler spread

(b) Taking into account the lattice periodic structure and repetitive struc-ture in time and the channel Doppler spread

Fig. 1. POPS-OFDM implementation methodology

shape durations are different will be quiet different as we

previously detailed for equal Tx/Rx pulse shapes durations.

In order to make the illustration tractable, we suppose that

the receiver waveform duration (Dψ) is greater than the

transmitter waveform duration (Dϕ), i.e Dψ > Dϕ.

1) ”Ping” step: For the kth iteration, we have ϕ(k−1)

available. We start by optimizing ψ according to

(34).

As is depicted in Fig.2(a), we calculate the

dispersion in the time according to the multipath

power profile. Then, we select the matrix used in

Formula.3 in Fig.3 to calculate the useful Kernel.

The size of the selection is quiet related to the

Dψ , since we are looking for the optimal receiver

waveform (ψ). After that, we shift the found matrix

according to the normalized symbol duration

N (See Fig.2(a)). Then, we select, as usual, the

matrix with the highest energy to be used in the

calculation of the interference Kernel matrix, based

on Formula.4 depicted in Fig.3. Finally, we calculate

the matrices which depend on the Doppler channel

and which will be computed once for both useful

and interference matrices in all the ”Ping” steps

(See Fig.3).

2) ”Pong” step: For the ”Pong” step, we have ψ(k)

available. First, we start by computing the temporel

inversion of the ψ(k) (ω(ψ(k−1))). Then based on

(15), we start by optimizing ω(ϕ) according to (24).

The ”Pong” step has the same approach as the

”Ping” step when we exchange the roles between

the ϕ and ω(ψ). Hence, as is depicted in Fig.3(a),

we calculate the dispersion in the time according

to the multipath power profile. Then, we select the

matrix used in Formula.5 in Fig.3(b) in order to

calculate the useful Kernel. The size of the section is

quiet related to the Dϕ, since we are looking for the

optimal receiver waveform (ω(ϕ)). After that, we

shift the found matrix according to the normalized

symbol duration N (See Fig.3(a)). Then, we reiterate

14

(a) Taking into account the lattice periodic structure and repetitive struc-ture in frequency and the channel Doppler spread

(b) Taking into account the lattice periodic structure and repetitivestructure in time and the channel Doppler spread

Fig. 2. POPS-OFDM implementation methodology (Dψ > Dϕ): ”Ping”step

the same previous reasoning: We select the matrix

with the highest energy to be used in the calculation

of the interference kernel matrix, based on For-

mula.6 depicted in Fig.3(b). Finally, we calculate the

matrices that depend on the Doppler channel and

which will be computed one time for both useful

and interference matrices in all the ”Pong” steps

(See Fig.3(b)). Note that once we have the optimized

ω(ϕ), we systematically deduce ϕ.

6.2 POPS-OFDM Performance

In Fig.4, we present the evolution of the SINR versus the

normalized maximum Doppler frequency for a normalized

channel delay spread values to BdF where Q = 128 and

for a waveform support duration equal to 3N . Through

this simulation, we determine the Doppler spread/ de-

lay spread balancing for a fixed channel spread value,

BdTm = 0.01. Also, we compare our optimized transmitter

waveform design with the conventional OFDM system that

deploys cyclic prefixes (CP) of 8 or 32 samples. This figure

demonstrates that our approach outperforms the conven-

tional OFDM system. Fig.5 shows the evolution of the SIR

in dB with respect to FT where we compare the POPS-

OFDM algorithm for different pulse shape durations with

the conventional OFDM algorithm. As can be expected, the

proposed system outperforms the conventional OFDM for a

large range of channel dispersions, especially in the case of

a highly frequency dispersive channel and this is whatever

the support duration of the waveform. This figure reveals

a significant increase that can reach 8dB in the obtained

SIR when the support duration increases. Furthermore, it

represents a mean to find the adequate couple (T, F ) of

an envisaged application to insure the desired transmission

quality. Then, we can note that for a lattice density equal

to δ = 0.8 (FT = 1.25), coinciding with a conventional

OFDM system with a CP having one quarter of the time

symbol duration, the SIR can be above 22dB for D = 1T .

Tx/Rx waveforms, mainly ϕopt and ψopt, corresponding to

the maximal SINR, are illustrated in Fig.6 for FT = 1.25,

BdTm = 0.01. This figure provides a comparison between

the optimized waveforms. As we remark in this figure, the

15

(a) Taking into account the lattice periodic structure and repetitive struc-ture in frequency and the channel Doppler spread

(b) Taking into account the lattice periodic structure and repetitive struc-ture in time and the channel Doppler spread

Fig. 3. POPS-OFDM implementation methodology (Dψ > Dϕ): ”Pong”step

(a) CP = N −Q = 8.

(b) CP = N −Q = 32.

Fig. 4. Doppler Spread-Delay Spread Balancing.

receiver and transmitter pulses are different from those of

the conventional OFDM system. This confirms our claims in

the sense that the conventional OFDM system not usually

lead to the optimal SINR. Fig.7 shows that the obtained

transmitter pulse reduces exponentially of about 80dB, the

out-of-band (OOB) emissions contrary to a conventional

OFDM system that requires large guard bands to do so and

it can be observed that the optimal prototype waveform

is more localized. We notice also that when D increases

the gain becomes less pronounced starting from a value of

D = 5T . More importantly, since the optimized obtained

waveform reduces dramatically the spectral leakage to

neighboring subcarriers, inter-user interference will be min-

16

Fig. 5. Performance and Gain in SIRdB-Identical Tx/Rx Pulse ShapeDurations.

imized especially at the uplink of OFDMA systems, where

users arrive at the base station with different powers. Fig.8

presents the evolution of the SIR in dB with respect to FT

where we compare the POPS-OFDM algorithm for different

Tx/Rx pulse shape durations with the conventional OFDM

algorithm for Q = 128 and for channel spread (BdTm)

equal to 0.01. We start by fixing the transmitter waveform

duration and increase gradually the receiver waveform

duration. As it is expected, when we increase the receiver

waveform duration (Dψ) in both cases: from Dψ = 1N to

Dψ = 3N (See Fig.6-(a)) and from Dψ = 3N to Dψ = 5N

(See Fig.6-(b)), the SIR is slightly superior to that obtained

when we maintain the same Tx/Rx waveform durations

(Dψ = Dϕ = 1N and Dψ = Dϕ = 3N , respectively).

We remark also that when we consider different values

of the receiver pulse shape duration which is different to

the transmitter pulse shape duration, we obtain the same

performance in terms of SIR. This behavior can be explained

by the radical change occurred at the first increase of the

receiver pulse duration. Hence, every increase after the first

modification where the Tx/Rx pulse shape durations are no

longer equal, the gain in terms of SIR is negligible (see Fig8).

(a) D = 5T .

(b) D = 7T .

Fig. 6. Tx/Rx Waveforms Optimization Results.

6.3 Dependency to waveforms initializations

Since POPS-OFDM banks on an iterative approach to find

the optimal waveform, it is wise to study the POPS-OFDM

performance in term of its sensitivity to different wave-

forms prototype initialization. Motivated by the fact that the

Hermite functions form an orthonormal base of the Hilbert

space L2(R) of square summable functions and offer in a

decreasing order the best localization in time and frequency,

we initiate POPS-OFDM with different linear combination

of 8 Hermite functions which are the most localized. Also,

we consider gaussian waveforms where we vary the mean

and the standard variation, in addition to the root-raised co-

sine initializations for different Roll-off factor. Fig.9 depicts

the existence of local maxima, but in the almost cases, POPS-

17

(a) Spectrum of One Subcarrier.

(b) Spectrum of 64 Subcarriers.

Fig. 7. Normalized Power Spectral Density (PSD) in dB.

OFDM is not trapped and converges for the same optimal

maxima. Cause of hardware computation limitation, we

calculate the global optimized SINR based on Algorithm 2

(see Section 5) for Q = 64 and D = 1T which is drawn in

Fig.7 − (b). Unfortunately, we conclude that whatever the

considered initializations, the optimized SINR is below the

global SINR target in this simulation context. But, in almost

cases, it outperforms the SINR offered by the conventional

OFDM system.

(a) Dϕ = 1N .

(b) Dϕ = 3N .

Fig. 8. Ventilation of Complexity Between Transmitter and Receiver.

6.4 Robustness characterization

As it is known, the synchronization is a crucial indicator for

efficiency of wireless communication systems and eventu-

ally for 5G [3], [4]. Generally, such systems are so sensible to

any synchronization error. As POPS-OFDM was principally

conceived to non-orthogonal future wireless multi-carrier, it

is recommended to evaluate its vulnerability against syn-

chronization errors.

In this section, we investigate the time and frequency syn-

chronization errors. Then, we focus on the sensibility of the

optimized waveforms for any variation around the optimal

BdTm.

In Fig.10, we can see clearly that the proposed algorithm

18

(a) Q = 128, CP = 32, D = 3T

(b) Q = 64, CP = 16, D = 1T

Fig. 9. Impact of Different Waveforms Prototype Initializations.

outperforms the conventional OFDM in terms of robustness

against the time synchronization errors when CP = 32 and

CP = 16. For the frequency synchronization errors, the

efficient proposed algorithm doesn’t degrade the SIR per-

formance compared to the conventional OFDM (See Fig.11).

Fig.12 illustrates the sensitivity of POPS-OFDM when we

assume a synchronization error on BdTm varying between

0.001 and 0.01. In this figure, we represent the SIR obtained

after optimizing the waveform when BdTm1= 0.001,

respectively when BdTm2=0.01. We remark that the SINR

performance of BdTm2degrades slowly comparing to that

where the optimization is realized forBdTm1. Therefore, it is

advantageous to optimize our system for large BdTm when

we do not know its optimal value.

Fig. 10. Sensitivity to Synchronization Errors in Time.

Fig. 11. Sensitivity to Synchronization Errors in Frequency.

Fig. 12. Sensitivity to an Estimation Error on BdTm.

19

7 CONCLUSION

In this paper, we investigated an optimal waveform design

for multicarrier transmissions over rapidly time-varying

and strongly delay-spread channels. For this purpose, a

novel optimizing algorithm for the transmitter and receiver

waveforms is proposed. The optimized waveforms provide

a neat reduction in ICI/ISI and guarantee maximal SINRs

for realistic mobile radio channels. In addition to that, POPS-

OFDM waveforms offer 6 orders of magnitude reduction

in out-of-band emissions and reveal a great robustness to

synchronization errors. Simulation results demonstrated the

excellent performance of the proposed solutions and high-

lighted the property of the efficient reduction of the spectral

leakage obtained through the optimized waveforms. To

test the robustness of the POPS algorithm, we evaluated

its sensitivity to time and frequency synchronization and

also to the initialization parameters. The obtained results

showed the good performance of our waveforms optimiza-

tion algorithm even in high mobility propagation channels.

As such, our proposed solutions can be seen as an attractive

candidate for the optimization of the spectrum allocation in

5G systems. A possible challenging research axis consists in

extending the optimization for the OQAM/OFDM systems.

Another interesting perspective can be investigated such

that the design of OFDM pulse shapes optimized for partial

equalization, for carrier aggregation and for a lower latency,

with tolerant to bursty communications with relaxed syn-

chronization.

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